I've heard before that Jesus was recorded on the census that is mentioned in Luke 2:1, and that his name was the last name recorded. I've also heard that Herod would then use this census when he ordered the killing of the children (Mat 2:16.)

Is there any proof that has survived on this, and if it did, are we able to see his name still recorded on the census?

3 Answers
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13 When they had gone, an angel of the Lord appeared to Joseph in a dream. “Get up,” he said, “take the child and his mother and escape to Egypt. Stay there until I tell you, for Herod is going to search for the child to kill him.”

14 So he got up, took the child and his mother during the night and left for Egypt, 15 where he stayed until the death of Herod. And so was fulfilled what the Lord had said through the prophet: “Out of Egypt I called my son.

My guess is that they probably weren't part of the "head count", but fled the city instead.

While there is a recorded history of the census (in the Bible), there's no indication that Jesus or his parents were counted.

This is actually a curious question. Wikipedia actually suggests the existence of the census is in doubt.

In Christianity, the Gospel of Luke connects the birth of Jesus to a "worldwide census" in which individuals had to return to their ancestral cities. Jesus' parents, Joseph and Mary, travel from their home in Nazareth, Galilee, to Bethlehem, where Jesus is born. This explains how Jesus, a Galilean, could have been born in Bethlehem in Judea, the city of King David. No other record of such a census exists, and the idea of everyone in the Roman Empire returning to an ancestral city for a census is questioned by scholars.[3][4] The Gospel of Matthew has a different birth narrative, with Jesus' birth taking place during the life of Herod the Great, who died c. 4 BC. Bible scholars have traditionally sought to reconcile these accounts; while most current scholars regard this as an error by the author of the Gospel of Luke,[5] thus casting doubt on the Historical reliability of the Gospels.

Thus there is obviously no physical record of the census documents. Therefore it would be impossible to know if Jesus was counted among them, or was the last one mentioned.

There is no surviving Roman census document that lists Jesus. But this isn't particularly surprising: most such ancient documents don't survive. It's not like we have a copy of the census rolls and they jump from "Jeremiah" to "Jethro" with Jesus glaringly omitted. The rolls of that census no longer exist.

We do have surviving copies of a book by Tertullian, "An Answer to the Jews", written ca AD 197, in which he makes a brief mention, when describing Jesus's background, "As, among the Romans, Mary is described in the census, of whom is born Christ." Similarly, in his book "Against Marcion", he mentions Jesus's "enrolment in the census of Augustus— that most faithful witness of the Lord's nativity, kept in the archives of Rome". He makes these statements in a very straightforward and confidant way -- no weasle words like "these records are probably there" or "they were there at one time". There's a pretty clear implication that anyone who wants to check up on his facts could do so. If he was just making this up, he was a great bluffer. He would have made an excellent poker player.

Tertullian was the son of a Roman centurion and there is some evidence that he himself was a lawyer, so he might well have seen the census documents himself or spoken to someone who had.